The problem I'm having.. is that the bit walks. If it hits resistance in the block, it shuffles over a quarter-inch or so. That wouldn't be a huge deal if it shifted a quarter inch *in the same direction* for all three holes -- but I'm not that lucky. I'd prefer not to end up an example of bad electrical practice by hanging this socket on a single screw...

The magic trick is chucking that masonry bit into a hammer drill. A regular drill has the tendency to do exactly what you're saying it does, instead of hammering the concrete into chunks the bit can then remove, it walks to the side.

I did a similar thing but to a poured concrete wall. Rather than fight with the drill bit moving around, I used a Hilti nail gun to anchor a piece of 3/4 inch plywood to the wall. I could then mount the dryer outlet to the plywood with ordinary wood screws.

Another way that might work is to use your current carbide drill bit and let the plastic anchor holes go wherever they want, then drill matching holes in the dryer bracket. At least one hole will line up.

DO NOT mount it directly to the concrete wall. YOU WILL REGRET IT, when the time comes to replace it. (Or the next home owner does. And Yes, it will need to be replaced. Typically those outlets do after 10 years or so.).

As suggested above. Mount a piece of wood to the wall, and mount the receptacle to the wood. A scrap piece of 2x6 or plywood, or anything secure.

Get a long enough piece of wood to affix the cable up to the studs where it runs horizontally.

Mounting the outlet to a board is generally acceptable.
Using a hammer drill with a TapCon screw anchor is the modern way, but you need to have a lot of elbow grease to keep the drill steady.
In the Old Days, I'd use a hardened punch and heavy hammer to start the first hole. Mount the back plate of the outlet with the one screw, then use the holes in the back plate as a guide when drilling the other holes. Remove back plate, insert the remaining anchors, then finish attaching. Put the first hole in the mortar between the blocks, avoiding the aggregate in the blocks makes drilling that hole easier.

You could mount the outlet to the floor joists above instead.
But then you'd have to mount the dryer to the wall...

KPJL FM wrote: ↑
Thu Jul 20, 2017 8:28 am
When you're all done, you'll find the dryer's cord is 6 inches to short to reach the new outlet.
Or as my wife would say, I tried to add an outlet, ended up replacing the roof.

With my luck, everything will fit perfectly except the plug on the end of the cord will be different than the outlet socket.

KPJL FM wrote: ↑
Thu Jul 20, 2017 8:28 am
When you're all done, you'll find the dryer's cord is 6 inches to short to reach the new outlet.
Or as my wife would say, I tried to add an outlet, ended up replacing the roof.

With my luck, everything will fit perfectly except the plug on the end of the cord will be different than the outlet socket.

KPJL FM wrote: ↑
Thu Jul 20, 2017 8:28 am
When you're all done, you'll find the dryer's cord is 6 inches to short to reach the new outlet.
Or as my wife would say, I tried to add an outlet, ended up replacing the roof.

With my luck, everything will fit perfectly except the plug on the end of the cord will be different than the outlet socket.

Fossil

Please don't say that:( (either of you)

Don't worry. The dryer cords are pretty long. And there are only two types of Dryer plugs. 3 Prong (NEMA 10-30R), or 4 Prong (NEMA 14-30R).
The picture you posted, the 30R, is the most common type. The dryer uses balanced 240, so there is no neutral needed. The 3rd (center) conductor is simply a ground. Some of the more electronic dryers (and Stoves too) have complex enough control systems that need 120V to run, so you have both a Neutral and Ground going to them. (My stove is a 4 prong, but the dryer remains a 3 prong.)

Since I mounted my dryer outlet on wood nailed to the poured concrete wall, here's another funny. When I bought my house in 1987, it came with a washer and dryer (still in service today and working great; the quality went in before the GE nameplate went on). There was a separate circuit run for the washing machine and the outlet was mounted up in the floor joists above the machine. The right-angle U-ground plug at the end of the washer's power cord had been cut off and a 3-pole twist-lock plug installed. This is because the electrician oriented the outlet such that the plug went in vertically and hung down, so the locking plug was needed. Why he didn't just put in an ordinary duplex outlet and mount it such that the original cord could hang down like it was designed to in a normal wall outlet is beyond me. Sometime after I had moved in, I found a right-angle U-ground plug with black flat wire lying on the floor, the exact plug that had been replaced by the twist-lock. I moved the twist-lock receptacle box to my wood plate on the wall that I used for the dryer outlet, installed an ordinary duplex outlet, and spliced (twisted, soldered, and well-taped) the original plug back onto the washing machine's cord, and plugged it where it belonged. Still not sure why it needed to be modified at all. Sometimes electricians do strange things.